A posteriori error control for the Allen–Cahn problem: circumventing Gronwall's inequality

Daniel Kessler; Ricardo H. Nochetto; Alfred Schmidt

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 38, Issue: 1, page 129-142
  • ISSN: 0764-583X

Abstract

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Phase-field models, the simplest of which is Allen–Cahn's problem, are characterized by a small parameter ε that dictates the interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on ε-2. Using an energy argument combined with a topological continuation argument and a spectral estimate, we establish an a posteriori error control result with only a low order polynomial dependence in ε-1. Our result is applicable to any conforming discretization technique that allows for a posteriori residual estimation. Residual estimators for an adaptive finite element scheme are derived to illustrate the theory.

How to cite

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Kessler, Daniel, Nochetto, Ricardo H., and Schmidt, Alfred. "A posteriori error control for the Allen–Cahn problem: circumventing Gronwall's inequality." ESAIM: Mathematical Modelling and Numerical Analysis 38.1 (2010): 129-142. <http://eudml.org/doc/194202>.

@article{Kessler2010,
abstract = { Phase-field models, the simplest of which is Allen–Cahn's problem, are characterized by a small parameter ε that dictates the interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on ε-2. Using an energy argument combined with a topological continuation argument and a spectral estimate, we establish an a posteriori error control result with only a low order polynomial dependence in ε-1. Our result is applicable to any conforming discretization technique that allows for a posteriori residual estimation. Residual estimators for an adaptive finite element scheme are derived to illustrate the theory. },
author = {Kessler, Daniel, Nochetto, Ricardo H., Schmidt, Alfred},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {A posteriori error estimates; phase-field models; adaptive finite element method.; Allen-Cahn equation; Phase-field models; mesh adaptation techniques; a posteriori error control; adaptive finite element},
language = {eng},
month = {3},
number = {1},
pages = {129-142},
publisher = {EDP Sciences},
title = {A posteriori error control for the Allen–Cahn problem: circumventing Gronwall's inequality},
url = {http://eudml.org/doc/194202},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Kessler, Daniel
AU - Nochetto, Ricardo H.
AU - Schmidt, Alfred
TI - A posteriori error control for the Allen–Cahn problem: circumventing Gronwall's inequality
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 1
SP - 129
EP - 142
AB - Phase-field models, the simplest of which is Allen–Cahn's problem, are characterized by a small parameter ε that dictates the interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on ε-2. Using an energy argument combined with a topological continuation argument and a spectral estimate, we establish an a posteriori error control result with only a low order polynomial dependence in ε-1. Our result is applicable to any conforming discretization technique that allows for a posteriori residual estimation. Residual estimators for an adaptive finite element scheme are derived to illustrate the theory.
LA - eng
KW - A posteriori error estimates; phase-field models; adaptive finite element method.; Allen-Cahn equation; Phase-field models; mesh adaptation techniques; a posteriori error control; adaptive finite element
UR - http://eudml.org/doc/194202
ER -

References

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  9. X. Feng and A. Prohl, Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows. Num. Math.94 (2003) 33–65.  
  10. Ch. Makridakis and R.H. Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal.41 (2003) 1585–1594.  
  11. J. Rappaz and J.-F. Scheid, Existence of solutions to a phase-field model for the solidification process of a binary alloy. Math. Methods Appl. Sci.23 (2000) 491–513.  
  12. A. Schmidt and K. Siebert, ALBERT: An adaptive hierarchical finite element toolbox. Preprint 06/2000, Freiburg edition.  

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